Integral of 5x⁹ dx

1. The integral of a constant times a function is the constant times the integral of the function:

   $$\int 5 x^{9}\, dx = 5 \int x^{9}\, dx$$

   1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

      $$\int x^{9}\, dx = \frac{x^{10}}{10}$$

   So, the result is: $\frac{x^{10}}{2}$

2. Add the constant of integration:

   $$\frac{x^{10}}{2}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{10}}{2}+ \mathrm{constant}$$